Logic Notes

Hi there, thanks for coming to the thread about the single most widely abused word in all of internet discussion forumdom: logic.

A few months back, I decided to pick up a textbook on the subject and work through it cover to cover. I’m about halfway done, but I thought I’d present the notes I made here for discussion. I’d like to do something similar to what I am trying to do in my QFT thread in the Homework forum (and hopefully revive that thread, too). Anywho, the book is called Logic (duh) by Robert Baum, although you will not need a copy of the book to follow this thread. My notes are formatted like a geometry textbook, with definitions, axioms and theorems numbered. It is supposed to be a first undergraduate course in logic, so I presume that the only prerequisite is a high school education. I think that this is probably the most overdue topic in the philosophy forum, and maybe even all of PF in general.

Also, in just about every discussion in the Philosophy forum with which I get involved, it seems that it always boils down to a discussion of (and disagreement over) “what logic is”. Since I don’t have time to keep up with all those threads, I’m going to consolidate here.

Section 0.1: Defining Logic
Baum asserts that there is no single agreed upon definition of logic among logicians.

[?] Is logic the study of the laws of thought?
That’s too broad. There is plenty of thought that logic is not concerned with. For example, imagining two-headed goats is thought, but the logician does not care about that. We are interested in that particular subset of thought called reasoning

[?]Ah, so logic is the study of the laws of reasoning then, right?
Still too broad, and the root of the problem lies in the two uses of the word “law”. Laws can be either descriptive or prescriptive. A descriptive law is a statement of how something is done, as in the laws of nature. They cannot be broken or repealed. On the other hand, a prescriptive law is a statement of how something should be done, as in the laws set forth by a legislature. They can be broken and changed. The study of the descriptive laws of reason (how people do, in fact, reason) is not logic, but psychology. But the study of the prescriptive laws of reason (how people ought to reason) is logic.

[?]So, we are interested in the latter—the prescriptive laws of reasoning.
Correct. That will be our definition of logic throughout this study

[?]Baum does not say what any of those other definitions of logic are. Any of our resident philosophers have a clue as to alternative definitions?

Section 0.2: Recognizing an Argument

Sentences and Statements
One of the first subtleties here is the difference between a sentence and a statement. A sentence is a physical entity (ink on a page, pixels on a screen, or sound waves traveling through the air). It is also a linguistic entity, with letters and words organized according to grammatical rules. On the other hand, a statement is an abstract concept which is expressed by a sentence. (This point is, IMO, overly technical for this study. When I refer to a sentence as a statement, I’m going to ask you to just let it go, and keep it in mind that I am referring to “the statement expressed by that sentence”. Even Baum does this after explaining the sentence/statement distinction).

Definition 0.1: A statement is an idea about which it makes sense to say that it is true or false. It is synonymous with proposition.

While it is true that all statements are expressible by sentences, it is not the case the every sentence expresses a statement. It makes no sense to speak of the truth of a question, command, or exclamation. Statements are those ideas that can always be represented by a declarative sentence.

Look at these examples:

1. John is sitting in a chair.
2. Did you go out today?
1. Take out the garbage.
1. Is France in Europe?
1. Didn’t I see you at the movie last night?
1. Well, duhhh!

Sentence 1 is a declarative sentence. We do not know whether or not John is sitting in a chair, but we do know this: the statement is definitely either true or false, and so it expresses a statement.

Sentence 2 is a question. It makes no sense to say that a question is true or false, and so this does not express a statement.

Sentence 3 is a command. This, too, cannot be said to be true or false, and so it does not express a statement.

Sentence 4 is also a question, but there’s a little more going on here than with sentence 2. If I ask you this question, the statement, “I do not know where France is” is implied. However, that is not the intention of the question, as the declarative version is not expressed in anticipation of an answer. Thus, care has to be taken not to alter meaning when translating plain language sentences into sentences that express statements. This is the part I am having the most trouble with.

Sentence 5 is another question, but can be translated directly into the declarative sentence, “I saw you at the movies last night.”, and thus expresses a statement.

Sentence 6 (I know it’s a stretch to call that a sentence) is an exclamation, but in a certain context can be interpreted as a statement. If it is uttered in response to a question, “Check out that chick, isn’t she hot?”, then sentence 6 could be interpreted to mean the same thing as the declarative sentence, “That chick is hot.” Thus, we see that context can be important in translating plain language sentences into sentences that express statements.

Moral: All statements can be expressed by declarative sentences.

Argument Defined
This is where we start to really formalize our language. Some words are going to be given definitions which differ from those of their everyday usage, so it’s important to note and keep track of them.

Definition 0.2: An argument is a collection of statements, one of which is designated the conclusion and the remainder of which are premises asserted in support of that conclusion.

By this definition then, every argument contains exactly one conclusion and at least one premise.

Premises and Conclusions
By the Defintion 0.2, every argument contains exactly one conclusion and at least one premise. When analyzing arguments, identifying the conclusion and premises is the first order of business. This is usually facilitated by the use of certain indicator words, as illustrated in the arguments below (premises are in blue and conclusions are in red):

1. Since Jim has always voted Democrat in the past, he will probably do so again.
1. The negotiations will probably fail, because neither party is willing to compromise.
1. All mammals nurse their young; hence all elephants nurse their young, for all elephants are mammals.

This shows some examples of indicator words. A partial listing is given below.

Problems in Recognizing an Argument
Before analyzing an argument, it must be first recognized as such, and then translated into standard form (more on standard form in the following chapters). There are certain difficulties which may arise in this process.

Exposition vs. Argumentation
Consider the following:

The school was built in 1890. It is of red brick, with a columned portico in front. The main door and the first-floor windows are arched. A limestone cornice creates a horizontal line to balance the vertical thrust of the white columns.

The above is an example of exposition, which can sometimes be mistaken for argumentation (although probably not this simple example). The difference between exposition and argumentation is that the former has no conclusion, or point that the author is trying to sell. It is just a list of facts.

Complex Propositions
Another thing that is often mistaken for an argument is something like the following:

If you jump off a building, you will fall.

The above is not an argument, but a proposition. It just happens to be a complex proposition, namely a conditional proposition. Complex propositions are not arguments in themselves, but single premises (or even a conclusion). We will get into this in much more detail in Chapter 3: Truth Functional Propositions, of which the conditional is an example.

Misidentification of Indicator Words
Consider the following:

1. It is foggy and the roads are bad, so don’t try to drive today.
1. Since Mary turned 50, she has been depressed.

Both of the above contain our old friends, the “indicator words”. But are they arguments?

The first one has the word “so”, which we may be tempted to take as indicative of a conclusion. But what would that conclusion be? It would be “don’t try to drive today”, which is a command, and so does not express a statement. Case 1, therefore, is not an argument.

The second one has the word “since”, which we may be tempted to take as indicative of a premise. However, in this case it expresses a temporal, as opposed to logical, relation. The temporal use of the word “since” is not indicative of a premise, and thus Case 2 is also not an argument.

Missing Indicator Words

Not all arguments are expressed using the indicator words, nor are they required to be. Nuff said.

Definition 0.3: An enthymeme is an argument which has one or more premises left out.

Definition 0.4: A non-sequitir is an argument whose conclusion does not follow from its premises.

There will be much more on non-sequitirs in the following chapters.

Consider the following:

The sun is shining today.
Today is Wednesday.
Jim will get an A in calculus.

There does not appear to be any connection between those propositions whatsoever. Does the conclusion really not follow from the premises, or is this argument simply not stated fully? In other words, is it a non-sequitir or an enthymeme? If we supply missing premises as follows:

The sun is shining today.
Today is Wednesday.
[The final exam in calculus is on Wednesday.]
[The exam counts for 10 percent of the final grade.]
[Jim has a 90 average going into the exam.]
[Jim has always done well on sunny days.]
[The cutoff point for an A is 89.5 pecent.]
Therefore, Jim will get an A in calculus.

then the argument looks to be in much better shape.

Moral: Good arguments do not have any logical links missing. We will study this in great detail in Parts I and II of the book.

This brings us to a time honored tradition among philosophers called the principle of charity. Among other things, charity dictates that, in a debate, one should assist one’s opponent as much as possible by supplying additional premises when confronted with an enthymeme if at all possible.

One last thing….Inferences and Arguments

Definition 0.5:Inference is the psychological process of moving from one thought to another presumably related thought.

“Infer” is different from “imply”. The premises of an argument imply its conclusion, where as a person infers conclusions from given premises. It is only appropriate to say that people “infer” things.

Consider the following:

James saw his plants start to wilt and thought, “I need to water them.”

The above describes an inference, not an argument. The two are not the same thing. However, for every inference there exists a corresponding argument, and an inference is only as good as its corresponding argument. We’ll get to what a “good argument” is shortly.

The argument corresponding to the above inference is:

In most instances in the past, if my plants started to wilt, then they required water.
My plants are starting to wilt.
Therefore, probably my plants require water.

Now we’re touching on the substance of Parts I and II of this book. The most important concept in deductive logic is that of validity.

Definition 0.6: A valid argument is an argument whose premises necessarily imply its conclusion. Equivalently, an argument is valid if and only if its conclusion is completely contained in its premises. Equivalently again, a valid argument is an argument whose premises provide absolute support for its conclusion.

One consequence of deductive validity is that the truth of the conclusion is not affected by new information (provided the premises are known to be true). In other words, the addition of any new premises has no effect on a valid argument. Let’s take a look at some invalid arguments.

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

This is what is known as a syllogism, which is the subject of Chapter 2. We will learn how to analyze these arguments in detail at that time, but for now let’s just try to get a feel for validity. The first premise puts the entire class of men in the class of mortals. That means that any individual man must also be a mortal. The second premise takes one such individual and asserts that he is a man. Putting the two together, we have the conclusion. With a little thought, it should be apparent that the above argument is valid.

Now take a look at another one.

All silamons are wistacious.
Piliute is a silamon.
Therefore, piliute is wistacious.

The above argument contains nonsense words, but other than that it is identical to the Socrates syllogism above. The argument here is also deductively valid, because if all things in the class of silamons (whatever they are) is in the class of things that are wistacious (whatever that is), than any individual in the class of silamons must also be in the class of things that are wistacious.

Moral: Deductive validity is determined only by logical form, and not by the content of an argument.

Let’s look at one more.

All dogs have 5 legs.
Fido is a dog.
Therefore, Fido has 5 legs.

The conclusion of the above argument is false, obviously because of the first premise. And yet the argument is deductively valid (hey, it’s the same as the other two).

Moral: Deductive validity is no safeguard against false conclusions. In fact, as we shall see, the truth of the statements in an argument cannot be determined by logic, and so logicians typically restrict their attention to analysis of validity.

Counterexamples
Now let’s look at some invalid arguments, and one way to expose them: via counterexamples. Note that all of the statements in the argument are true, demonstrating once again that validity is not related to truth.

Some men are courageous.
Some men are considerate.
Therefore, some men are courageous and considerate.

[?]You said that this argument is invalid. How on Earth can you tell that?Remember the definition of validity: If the premises of a valid argument is true, then the conclusion must be true. So, one way to expose an invalid argument is to construct an argument of the same form whose premises are true and whose conclusion is false. That argument is called a counterexample.

The easiest way to construct a counterexample is to start by choosing a false conclusion of the same form as the conclusion of the argument. A good one would be:

Therefore, some animals are cats and dogs.

Now, carrying the terms to the premises we have:

Some animals are dogs. True
Some animals are cats. True
Therefore, some animals are dogs and cats. False

And so this argument cannot be valid.

[?]Hold on! Cats and dogs have nothing to do with courageous and considerate men. How can you call this a counterexample?!Don’t forget what we’re doing here. We are analyzing the logical form of the argument. When we are checking validity, we don’t care about the content. If a single instance of true premises leading to a false conclusion is found, then we have exposed the argument as invalid.
[?]That seems difficult. What happens if you can’t think of a counterexample or if an argument is too compicated to do such a thing?That’s a good point. Constructing counterexamples is not the most general way of exposing invalid syllogisms. Soon we will learn to analyze arguments with Venn diagrams and truth tables, and our study of deductive logic will culminate with an introduction to formal proof. Once we are finished with parts I and II, we will have constructed a mechanical decision procedure (you’ll be seeing those three words a lot in the future) to analyze arguments. That mechanical decision procedure is logic.
[?]Ah, so if the mechanical decision procedure is logic, and logic is the set of prescriptive laws of reasoning, then the mechanical decision procedure must be those laws!That is not only correct, it is also a valid argument.

There has to be a way to characterize an argument as either “good” or “bad”. Earlier we saw a valid argument with a false conclusion (“5 legged Fido”) and we saw an invalid argument with a true conclusion (“courageous considerate men”). Surely, neither of these could be called “good” arguments, if the argument is meant to convince someone of something. In order for an argument to be considered good, it must:

1. Be correctly reasoned (IOW, it must be valid), as in the case of the first of those two arguments.
1. Have true premises.

Putting 1 and 2 together leads to the corollary that the argument must have a true conclusion, by the definition of valid.

Are the above criteria enough? Consider the following argument:

All men are mortal.
Socrates is a man.
Therefore, Socrates is a man.

The methods we will learn in subsequent chapters will not expose this as invalid, so criterion 1 is met. Also, the premises (and thus conclusion) are true, so criterion 2 is met. But is this argument “good”? Hardly, because the conclusion is simply one of the premises. This brings us to the concept of circularity in arguments.

Definition 0.7: An argument is circular if the conclusion is merely a restatement of one of the premises.

In a sense, every valid argument is circular because the conclusion must be completely contained in the premises. However, we object to this particular case because the conclusion is trivially true if the corresponding premise is true.

Now we know how to judge whether an argument is good or bad. Logicians call such arguments “sound” and “unsound”, respectively.

Definition 0.8: An argument is sound if it is valid, noncircular, and if the truth of its premises is well established.

As noted earlier, the business of determining truth is outside the scope of logic. Logicians rarely concern themselves with the soundness of an argument. Their expertise lies in determining validity.

We defined an argument as a collection of statements, one of which is the conclusion and the rest of which are premises offered in support of that conclusion. In the last section we looked at those arguments whose premises provide absolute support for the conclusion. That is not the only kind of argument.

This kind of argumentation that stems from observation is called inductive argumentation. Absolute terms such as “valid” and “sound” do not make any sense when applied to inductive arguments, so we will not use them for that purpose.

Definition 0.9: Any argument that is not deductively valid is an inductive argument.

Note a couple of differences between deduction and induction:

1. The premises of deductive arguments provide absolute support for their conclusions, while the premises of inductive arguments provide probable support for their conclusions.
1. Deductive arguments are neither strengthened nor weakened by new premises, but inductive arguments are strengthened by new confirmatory premises and are completely overturned by a single counterinstance.

It should be noted, however, that very often the premises of deductive arguments are conclusions of inductive arguments (especially in science). We will get into the scientific method in Part III: Induction.

Good vs. Bad Inductive Arguments

Inductive arguments are neither “valid” nor “invalid”. Rather, we say that they become “stronger” or “weaker” with the addition of more premises.

I don’t want to get too involved in this now, because we will get to it in Part III.

Either later today or sometime tomorrow I’ll put up some sample arguments, and anyone who cares to can have a crack at analyzing them.

Step 1. Identify the premises and the conclusion.
Step 2. Supply missing premises, if necessary and if possible.
Step 3. Determine whether the argument is deductive or inductive.

I really should include a “Step 0: Make sure you are analyzing an argument and not an expository piece”, but let’s just assume you have already done that.

Identifiying Premises and Conclusions

We already had some introduction to this in Section 0.2: Recognizing an Argument. It was noted that premises and conclusions are often signaled by indicator words. But what if they are not so signaled? Well, we also have Definition 0.2 to go on: premises are offered in support of the conclusion.

Consider the following examples:

Argument 1:
Frank could never become a policeman. He is only 5’2’’ tall and weighs only 120 lbs.

Note that there are no indicator words here. Following the suggestion above, let us rely on Definition 0.2. I call the following method “The Why? Test.” Rearrange the argument as follows:

1. Frank could never become a policeman.
1. Frank is only 5’ 2’’ tall.
1. Frank weighs only 120 lbs.

The conclusion of this argument is rather obvious, but let us pretend that we do not see it. Let us say that we think that Statement 2 is the conclusion. According to Definition 0.2, the premises are purported to explain why the conclusion is true. So, we should be able to complete the following blank with either of the other two statements and get something sensible:

Frank is only 5’2’’ tall. Why? Because _________________.

Let’s try it.

Frank is only 5’2’’ tall. Why? Because Frank could never become a policeman.
Frank is only 5’2’’ tall. Why? Because Frank weighs only 120 lbs.

Looks like we misidentified the conclusion. Obviously, it is Statement 1.

Consider another example, the “swan” argument again (only slightly modified):

Argument 2:
1. Swan A was observed to be white.
2. Swan B was observed to be white.
3. Swan C was observed to be white.
4. Swan D was observed to be white.
5. All swans are white.

Again there are no indicator words, but applying The Why? Test should convince you that Statement 5 is the conclusion.

Supplying Missing Premises
No logical links can be missing in a valid deductive argument. When a collection of statements is identified as an argument and it is determined that the argument is not deductively valid, the reader should add premises to make the argument as good as possible. This is an extension of the Principle of Charity introduced in Section 0.2.

[?]What does that mean, “Make the argument as good as possible”?It means that one should fill in the missing premises required to make the argument sound, if possible.
[?]And what if it is not possible?If an argument simply cannot be made into a sound deductive argument then one should try to add premises such that the argument becomes a strong inductive argument. As we will see shortly, the Principle of Charity cannot always be successfully applied.

Let us go back to Argument 1. Clearly, it is incomplete as no link between vital statistics and qualification as a policeman was asserted. Can we add premises to make the argument deductively valid? Indeed we can.

Argument 1a:
1. Frank is only 5’2’’ tall.
2. Frank weighs only 120 lbs.
3. [The minimum height for police officers is 5’8’’.]
4. [Frank will never grow to be any taller.]
5. Frank could never become a policeman.

The argument is now deductively valid, although the added premises are subject to challenge. However, logicians are not so much interested in truth as they are in validity.

Now let us go back to Argument 2. Try as you might, you will not be able to make this argument deductively valid, because in order to do so you would have to collect data on every swan that exists (unless, of course, you found one that is not white). Since this argument is not deductively valid, it is by definition an inductive argument, and inductive arguments can be made stronger or weaker with new information. To make this argument stronger, we would have to collect information and add premises as follows:

Argument 2a:
1. Swan A was observed to be white.
2. Swan B was observed to be white.
3. Swan C was observed to be white.
4. Swan D was observed to be white.
5. Swan E was observed to be white.
6. Swan F was observed to be white.
7. All swans are white.

Determining Whether Arguments Are Deductive or Inductive
Do the premises necessarily imply the conclusion? If so, then you have a valid deductive argument in front of you. If not, then by Definition 0.9 the argument is inductive. I realize that we have not yet developed the systems of deductive and inductive logic, but the arguments we are looking at here are simple enough to analyze by inspection.

In Argument 1a, for example, it should not be difficult for you to convince yourself that, if the premises are true, then there is no way for the conclusion to be false. Thus, the argument is a valid deductive one.

In Argument 2a, on the other hand, the premises do not necessarily imply the conclusion. They lend only probable support (as opposed to absolute support) to the conclusion. Consequently, the argument can be strengthened by the addition of premises that support the conclusion, and completely overturned by a single counterinstance, for example, “Swan G is black.” Argument 2a is thus seen to be inductive.

Let us look at some more arguments, concentrating on adding premises to make the argument as good as possible, noting some peculiarities along the way. As we have already noted, this is the Principle of Charity.

The first thing to note is that the means by which an argument can be completed is not unique.

Look at the following argument.

Argument 3:
Kuwait has no right to control a large portion of such an essential resource as oil, since it has only a minute portion of the world’s population.

This argument is not valid as written, so let us add premises to make it so.

Argument 3a:
1. [No country which has only a small portion of the world’s population has the right to control a large potion of an essential natural resource.]
2. Kuwait has only a minute portion of the world’s population.
3. [Oil is an essential natural resource.]
4. Therefore, Kuwait has no right to control a large portion of such an essential resource as oil.

Now, the argument is valid, although Premise 1 is open to challenge. We could also have completed the argument as follows:

Argument 3b:
1. [If Kuwait has a small portion of the world’s population, it should not have the right to control a large portion of essential natural resource such as oil.]
2. Kuwait has a minute portion of the world’s population.
3. Therefore, Kuwait has no right to control a large portion of such an essential resource as oil.

Both interpretations are equally faithful to the intent of the original argument, which is what is important.

The second thing to note is that when one adds premises to similar arguments, one should be consistent about the form in which the premises are added.

Argument 4:
The United States has only 5 percent of the world’s population, so it does not have any right to consume 50 percent of the world’s natural resources.

The similarity between this and the ‘Kuwait’ argument is obvious. We can complete this argument in each of the following ways:

Argument 4a:
1. [No country which has only 5 percent of the world’s population has the right to consume 50 percent of anything in the world.]
2. [Natural resources are something in the world.]
3. The United States has only 5 percent of the world’s population.
4. Therefore, the United States does not have the right to consume 50 percent of the world’s natural resources.

or

Argument 4b:
1. [If a country has only 5 percent of the world’s population, it does not have the right to consume 50 percent of the world’s natural resources.]
2. The United States has 5 percent of the world’s population.
3. Therefore, the United States does not have the right to consume 50 percent of the world’s natural resources.

The thing to note here is that if argument 3a is used, then 4a should also be used, and likewise for 3b and 4b. This is not a point of logic, but of style.

The third thing to note is that sometimes an argument cannot be made into a sound deductive argument, but it can be made into a strong inductive argument.

Argument 5:
To reduce the high absenteeism rate among workers at our factory, we suggest that the work day be divided into three shifts of four hours each. Each worker would be required to work ten shifts each week, but he would be free to choose which shifts he will work. Such a system has been instituted in three German factories, and absenteeism in each of them declined significantly.

There are no indicator words, but applying “The Why? Test” should convince you that the conclusion is the change in work schedule will bring about the decline in absenteeism. Clearly, the argument is not deductively valid. Can we help make it better? Let’s see.

Argument 5a:
1. In German factory 1, the work day was divided into three four-hour shifts, each worker was free to choose which ten hour shifts he would work each week, and absenteeism declined significantly.
2. In German factory 2, the work day was divided into three four-hour shifts, each worker was free to choose which ten hour shifts he would work each week, and absenteeism declined significantly.
3. In German factory 3, the work day was divided into three four-hour shifts, each worker was free to choose which ten hour shifts he would work each week, and absenteeism declined significantly.
4. [In German factory 4, the work day was divided into three four-hour shifts, each worker was free to choose which ten hour shifts he would work each week, and absenteeism declined significantly.]
5. [In German factory 5, the work day was divided into three four-hour shifts, each worker was free to choose which ten hour shifts he would work each week, and absenteeism declined significantly.]
6. In our factory, the work day will be divided into three four-hour shifts, and each worker will be free to choose which ten shifts he will work each week.
7. Therefore, absenteeism in our factory will decline significantly.

The argument is still not deductive, but with the addition of two positive instances of the plan working in German factories, the argument becomes a stronger inductive argument. Of course, the necessary research would have to be done prior to asserting those statements.

Let’s try to make the argument deductively valid.

Argument 5b:
1. [If in German factories, the work day was divided into three four-hour shifts with each worker
free to choose which ten shifts a week he would work and absenteeism declined, then if we institute the same system, absenteeism will decline in our factory.]
2. In three German factories the work day was divided into three four-hour shifts with each worker free to choose which ten shifts a week he would work, and absenteeism declined.
3. Therefore, if we institute the same system, absenteeism will decline in our factory.

The argument is now deductively valid, but the first premise is highly questionable. In the spirit of Charity, one would probably feel compelled to go with the strong inductive argument, as opposed to a valid deductive argument which is almost certainly not sound.

The fourth and final thing to note is that Charity cannot always successfully be applied.

Consider the following argument.

Argument 6:
Shakespeare was an Englishman; consequently, all Englishmen are great playwrights.

What premises can we add to this to make it a good argument? A first try might be the following.

Argument 6a:
1. Shakespeare was an Englishman.
2. [Shakespeare was a great playwright.]
3. Consequently, all Englishmen are great playwrights.

The truth of the added premise is certainly well established (although there might be some disagreement). However, that premise does not make the argument deductively valid. If one plays around with this argument, one will see that there is basically nothing that can be done with this argument.

Some arguments are so poorly constructed that they are beyond the reach of even Charity.

Section 1.1: Categorical Statements
Categorical statements are the building blocks of categorical syllogistic logic, which is the subject of Part I (Chapters 1 and 2) of this book. This particular chapter is a little dry, so bear with me until we get to Chapter 2: Categorical Syllogisms, in which we will analyze arguments.

Some examples of categorical statements are given below.

All students are intelligent.
No humans are gorillas.
Some persons in this room are politicians.

These statements have two things in common, and those two things are what make these statements categorical.

1. The subject and predicate of each statement refers to a class (aka a category) of things. For example, in the first statement, the subject refers to the class of students, and the predicate refers to the class of intelligent persons.
2. Each statement has a quantifier. The quantifiers in each statement are, respectively, all, no, and some.
3. Each statement is either inclusive or exclusive, in that the subject class is either included in or excluded from the predicate class.

For definiteness:

Definition 1.1: A categorical statement asserts both a quantitative and qualitative relationship between its subject class and its predicate class.

Definition 1.2: The quality of a categorical statement is the characteristic that all or some of the members are either included in or excluded from the predicate class. Categorical statements of the former type are said to be affirmative and those of the latter type are said to be negative.

Definition 1.3: The quantity of a categorical statement is the part of the subject class that is either included in or excluded from the predicate class. Categorical statements that refer to all or none of the subject class are said to be universal, while those that refer to only some of the subject class are said to be particular.

As might be expected, the only 3 quantifiers of any use are all, some and no. Other possibilities, such as most, all but one, etc. are not usable as they do not easily lend themselves to formal analysis when used in syllogisms. This shows a clear limitation of syllogistic logic in dealing with quantitative problems. This shortcoming will be strongly remedied in Chapter 5: Quantitative Logic. Stay tuned…

Special Notes:
1. The quantifier some is taken to mean at least one and does not exclude all.
2. Statements whose subject class refers to one individual or thing are called singular and are treated as universal statements. Example: London is not a nice place to live can, for the purposes of syllogistic logic, be treated as “All of the class of things of which London is the only member is not in the class of nice places to live.”

Section 1.2: Types of Categorical Statements
There are exactly 4, which you may have already deduced. After all,
(2 quantifiers)*(2 quality states)=4 types of statements

I say 2 quantifiers because, when quality is taken into account, both of the universal quantifiers can be thought of as one.

The 4 types are listed below, and the shorthand name for each is given:
1. Universal Affirmative, aka an A statement. Example: All cats are carnivorous animals.
2. Particular Affirmative, aka an I statement. Example: Some university students live at home.
3. Universal Negative, aka an E statement. Example: No mammals have feathers.
4. Particular Negative, aka an O statement. Example: Some vegetarians do not eat eggs.

Standard Form:
In order to proceed with formal analysis of syllogisms, the statements must be put into standard form.

Definition 1.4: A statement is in standard form if it is of the form:
(Quantifier)(Subject Class)(Copula)(Predicate Class). The Copula is a form of the verb “to be”, and the Predicate Class should be a noun or noun clause.

To save time and space, one often finds it convenient to abbreviate categorical statements. This is done by choosing a representative letter from both the subject class and predicate class and letting those letters stand for their respective classes. For instance:

Statement: All required courses are introductory surveys.
Abbreviation: All C are S.

Statement: All cats are canrivorous.
Abbreviation: All C are V.

Two points.
1. A dictionary should be provided when abbreviating. Underlining suffices in this regard.
2. Never use the same letter to represent two different things. This is illustrated with the second of the two examples above.
3. It is customary to use capitalize the abbreviation, so as not to confuse them with logical variables, to be introduced next.

Schema:
The schema of a statement contains its logical form only. It substitutes variables, typically chosen from the middle of the alphabet (p,q,r,…) and written in lower case, for the subject and predicate classes. These forms do not represent statements, as it makes no sense to speak of their truth or falsehood.

The schema for each of the 4 types of categorical statement will now be discussed in turn. In the discussion, S and P stand for Subject and Predicate classes, respectively. I am including both the standard form and the schema for each statement. This may look redundant at first, but the reason is to drive the point home that the two are not the same. Abbreviated statements have definite truth values, and schema do not.

A Statements:
Standard Form: All S are P.
Schema: All p are q.

Special Notes:
1. Exclusive statements written in ordinary language which use terms such as “only” or “none but” are typically into standard form as A statements. Example: “Only friends are invited to my party” is translated into standard form as “All persons invited to my party are friends.” Note that the subject and predicate classes are reversed in standard form.
2. Sentences that use “the only” (as opposed to simply “only”) are not exclusive, and thus no reversal of subject and predicate classes is required. Example: “The only people invited to my party are friends” is translated into standard form as “All people invited to my party are friends.”
3. As mentioned previously, singular affirmative statements have subject classes with only one member. They are treated as A statements. Example: “Prudence believes in ESP” is translated into standard form as “The class of people of which Prudence is the only member is in the class of persons who believe in ESP.” This can be abbreviated as “All P are E”, or simply “P is E”.

E Statements:
Standard Form: No S are P.
Schema: No p are q.

Special Notes:
1. The Principle of Charity demands that one be faithful to the intent of the speaker when translating plain-language statements into standard form. Some consideration must be paid to the context.

Example: Employees who work here are not commuters.

Should this be translated as an E statement:

No E are C.

or as an O statement?

Some E are not C.

The intent of the reader is probably the latter, and indeed one should in general choose the weaker of several possible translations when the context provides no clues. This is done so as to not commit the speaker to too strong a position.

2. Singular negative statements are translated as E statements. Example: “Tom does not believe in ghosts” can be abbreviated as “No T are G.”

Special Notes:
1. Example: “Students were among those joining in the festivities” is translated into standard form as “Some S are F.” Note that there was no quantifier in the original statement.
2. There are other possible interpretations of the above example. Another is “Some non-students are people who joined in the festivities” and yet another is the conjunction of the two. Context should be looked to for clues.
3. Issues of charity arise here, too. Example: “Logic students argue with facility” could be taken to be either an A statement or an I statement. The weaker interpretation is to be chosen in the absence of context clues.
4. Loss of meaning in translation arises when quantifiers that are used in plain language that do not conform precisely to those of syllogistic logic. As noted previously, the only quantifiers used in standard form statements are the universal and particular ones we already noted. This is not so for plain-language statements. Consider the following.

Many Americans earn less than $5000 a year.
Most families own a TV.

These use as quantifiers many and most, which our syllogistic logic is ill-equipped to handle. There do exist other systems of logic that can handle them, but for now we will translate them into standard form as I statements.

5. Exceptive statements are statements that use quantifiers such as not quite all and almost
all, and they are translated into standard form as compound propositions. Consider the following.

Almost all of the students were at the game.

This is translated into the conjunction of I and O statements: (Some S are G) and
(Some S are not G).

All but freshmen students are eligible for the scholarship.

This is translated into the conjunction of A and E statements: (All non-F are S) and (No F are S).

Do not confuse exclusive with exceptive

O Statements:
Standard Form: Some S are not P.
Schema: Some p are not q.

Special Notes:
1. Example: “All of the students in the class are not present” can be interpreted in two ways: Either “No S are P” or “Some S are not P”. If context provides no clues, one should choose the second interpretation, which is weaker.

Segue: The Aristotelian Interpretation
The Aristotelian interpretation of stipulates that categorical statements carry existential import, meaning that the class in the subject term are stipulated to contain at least one member. Thus, in Aristotelian logic, statements such as

All mermaids have tails.

are not considered, because the class of mermaids has no members.

[?]So what? Can’t we just work with the statement, even though we know mermaids are make-believe?Ah, but we can also run into statements such asAll students who cut more than five class sessions will fail the course.
It could very well be that there are no students who cut more than five class sessions. This will prove to be problematic when we get to Immediate Inferences on the Aristotelian Interpretation.

We will first examine categorical statements under the Aristotelian interpretation, and then we will relax the assumption of existential import and reexamine those statements.

Section 1.3: Venn Diagrams on the Aristotelian Interpretation
We already showed one way to display the logical form of a categorical statement: the schema. We now show another way to do it: the Venn diagram.

What is a Venn diagram?
I found a website, http://www.venndiagram.com, but I cannot figure out how to generate the diagrams. In the meantime, we will have to do without having the diagrams in the notes. Once I correct the situation, I will edit this post to contain the diagram.

For now, I will have to go with a verbal description. An example of a Venn diagram is here. Categorical statements are diagrammed on the type on the right (2 circles), whereas categorical syllogisms are diagrammed on the type on the left (3 circles).

Rules for diagramming categorical statements:
1. Two overlapping circles are used to represent the classes of things referred to by the subject and predicate terms of the categorical statement. The circles should be labeled, according to the subject and predicate, for ease of use.
2. Any class that contains no members is to be shaded.
3. Any class that contains at least one member is to contain an X.
4. If it is not clear which class contains at least one member, but it clear that one or both of them must, then the X is placed on the line dividing the two classes.
5. The circles representing the subject and predicate classes are typically placed in a rectangle which represents the universe of discourse.
6. Under the Aristotelian interpretation, the subject class must have an X in some part of it (either the outside part, or its intersection with the predicate class).

Here are the basic procedures for diagramming the various categorical statements:

A Statements:All S are P.

You can infer from this statement that there are no S that are not P, so according to Rule 2 you can shade in the outside part of the circle representing the subject class. Then, according to Rule 3, you would place an X in the intersection of S and P.

E Statements:No S are P.

You can infer from this statement that there are no S that are in P, so according to Rule 2 you can shade in the intersection of the circles for S and P. Then, according to Rule 3, you would place an X in the outside part of the circle for S.

I Statements:Some S are P.

It is not determinable from this statement if any part of any class has no members (remember, ”some” means ”at least one” and so could mean ”all”). But we do know that S must have at least one member, and that that one must be in P, so by Rule 3 we place an X in the intersection of S and P. There is no shading.

O Statements:Some S are not P.

This is similar to the I Statement in that we cannot do any shading (why?), and that the statement tells us where to put the X. The X goes in the part of the circle for S not shared by that of P.

[?]Why are we doing this? You said that this is a way to display the logical form of a categorical statement, but we already have a way to do that: the schema. Why do this as well?Good question, and I have a good answer. Shortly, we will become interested in the notion of logical equivalence among categorical statements. As we will see, categorical statements are logically equivalent if and only if they have the same Venn diagram, despite the fact that they have different schemata.

It is important to have drawn Venn diagrams for each type of categorical statement before proceeding.

Definition 1.5: An immediate inference is a conclusion drawn from exactly one premise.

It is possible to make immediate inferences from each of the four types of categorical statement. There exist general rules for determining which of these are valid and which are not, and we will develop those rules here.

ContradictionDefinition 1.6: Two propositions are contradictory if the truth of one implies the falsehood of the other, and if the falsehood of one implies the truth of the other. That is, they are contradictory if they cannot have the same truth value.

Proposition 1.7:A statements and O statements with the same subject and predicate terms are contradictory.

Proposition 1.8:E statements and I statements with the same subject and predicate terms are contradictory.

The proofs of Propositions 1.7 and 1.8 are left as exercises, and will be supplied after you have had a chance to do them.

Examples:

Example 1:
All mammals are warm-blooded. (A statement)
Some mammals are not warm-blooded. (O statement)

ContrarietyDefinition 1.9: Two propositions are contrary if they cannot both be true, but they can both be false.

Proposition 1.10:A statements and E statements with the same subject and predicate terms are contrary.

Again, the proof is left as an exercise and will be supplied later.

Example:

Example 3:
All students are logicians. (A statement)
No students are logicians. (E statement)

Convince yourself that these are contrary.

At this point, you may be prompted to ask if a contrary relationship holds between A and O statements. It does not. Consider the following:

Example:

Example 4:
Some students are logicians. (I statement)
Some students are not logicians. (O statement)

Convince yourself that these can both be true, and are thus not contrary.

Is there a relationship that can be deduced between I and O statements? Indeed there is: subcontrariety.

SubcontrarietyDefinition 1.11: Two propositions are subcontraries if both cannot be false, but both can be true.

And, you guessed it…

Proposition 1.12:I statements and O statements with the same subject and predicate are subcontraries.

Again, the proof is left as an exercise and will be supplied later.

As an illustration, look at Example 4 again and convince yourself that the I and O statements are indeed subcontrary.

[?]So does that mean that E and I statements are subcontraries? I mean, if I is false, then E must be true, right? And wouldn’t the same for A and O?You’ve almost got it, but remember the full definition of subcontrariety. Neither A and O nor E and I can both be true, which is why they are not subcontraries. However, there are indeed relationships between them, namely subimplication and superimplication, and to these we now turn.